Equations of State in Engineering and Research - American Chemical

1 Practical Calculations of the Equation of State of Fluids and Fluid Mixtures Using Downloaded by NORTH CAROLINA STATE UNIV on August 22, 2013 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch001

Perturbation Theory and Related Theories DOUGLAS HENDERSON IBM Research Laboratory, San Jose, CA 95193

The properties of a fluid are determined largely by shortrange repulsive forces. The long-range attractive forces can be considered to be perturbations. Using these concepts, a perturbation theory of fluids is developed. In addition, the relationship of empirical equations of state to the perturba tion theory is examined. The major weakness of most empirical equations is the use of the van der Waals freevolume term, (V-Nb) , to represent the contributions of the repulsive forces. Replacement of this term by more satisfac tory expressions results in better agreement with experiment. -1

n p h e r e q u i r e m e n t s o f a n e q u a t i o n of state f r o m a t h e o r e t i c a l c h e m i s t a n d a c h e m i c a l engineer a r e s o m e w h a t

different.

T h e theoretical

c h e m i s t desires to u n d e r s t a n d t h e o r i g i n of t h e p r o p e r t i e s of t h e fluid h e is s t u d y i n g a n d often is less i n t e r e s t e d i n o b t a i n i n g h i g h l y a c c u r a t e agreement w i t h experimental data.

O n t h e other h a n d , t h e c h e m i c a l

e n g i n e e r w a n t s a s i m p l e , e m p i r i c a l e q u a t i o n of state w h i c h is i n close agreement w i t h e x p e r i m e n t a l d a t a . T h e q u e s t i o n of w h e t h e r this e m p i r i c a l e q u a t i o n o f state has a n y t h e o r e t i c a l basis is less i n t e r e s t i n g . B e c a u s e of this a p p a r e n t d i v e r g e n c e o f interests a n d needs, t h e r e has b e e n little i n t e r a c t i o n b e t w e e n

t h e o r e t i c a l chemists a n d c h e m i c a l

engineers w o r k i n g o n t h e e q u a t i o n of state o f fluids. T h i s is u n f o r t u n a t e b e c a u s e t h e r e q u i r e m e n t s o f t h e t w o g r o u p s are c o m p a t i b l e .

N o theore-

t i c i a n w o u l d c l a i m to u n d e r s t a n d f u l l y some p h e n o m e n a i f h e c o u l d n o t o b t a i n reasonable q u a n t i t a t i v e a g r e e m e n t w i t h e x p e r i m e n t . O n t h e o t h e r h a n d , a n e m p i r i c a l e q u a t i o n of state w i t h a w e a k o r e v e n f a u l t y t h e o r e t i c a l 0-8412-0500-0/79/33-182-001$07.50/l © 1979 American Chemical Society

In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.

2

EQUATIONS

O F

STATE

basis is no m o r e t h a n a n i n t e r p o l a t i o n s c h e m e a n d is q u i t e useless

for

e x t r a p o l a t i o n to t h e r m o d y n a m i c states for w h i c h e x p e r i m e n t a l d a t a are n o t a v a i l a b l e . P r e s u m a b l y c h e m i c a l engineers w o u l d p r e f e r to h a v e some p r e d i c t i v e c a p a b i l i t y a n d if the results of t h e t h e o r e t i c i a n c a n be expressed i n some u s e f u l f o r m w h i c h is c o n v e n i e n t for q u i c k c a l c u l a t i o n , c h e m i c a l engineers w o u l d b e i n t e r e s t e d . I assume that this gap b e t w e e n t h e o r e t i c a l chemists a n d c h e m i c a l engineers exists because, u n t i l v e r y r e c e n t l y , theoreticians h a d l i t t l e to Downloaded by NORTH CAROLINA STATE UNIV on August 22, 2013 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch001

offer c o n c e r n i n g the t h e o r y of

fluids.

H o w e v e r , d u r i n g the past d e c a d e

r a p i d progress has b e e n m a d e i n this area a n d a n a t t e m p t to b r i d g e t h i s g a p n o w seems a p p r o p r i a t e .

T h i s is a n a m b i t i o u s task a n d g i v e n the

d e a d l i n e s w h i c h are a n u n a v o i d a b l e p a r t of a n y conference,

i t is not a

task that I w o u l d c l a i m to a c c o m p l i s h f u l l y here. H o w e v e r , I h o p e that this c h a p t e r w i l l c o n t r i b u t e to the b r i d g i n g of this g a p . I s h a l l a t t e m p t to s u r v e y recent progress fluids.

i n t h e t h e o r y of

I w i l l p r o v i d e references to a l l of the m a j o r t e c h n i q u e s .

dense

However,

I w i l l e m p h a s i z e p e r t u r b a t i o n t h e o r y b e c a u s e I f e e l t h a t this is the t e c h n i q u e w h i c h is most i n t e r e s t i n g to c h e m i c a l engineers.

Further, I

w i l l s h o w that the p e r t u r b a t i b n t h e o r y c a n b e u s e d i n p a r t to justify c o m m o n e m p i r i c a l equations of state. M a n y of these equations are w e l l f o u n d e d i n theory.

H o w e v e r , w e s h a l l see t h a t there is one t e r m w h i c h

seems to a p p e a r i n a l l e m p i r i c a l e q u a t i o n s

of

state.

This term

has

a b s o l u t e l y no t h e o r e t i c a l basis a n d i t s h o u l d be d i s c a r d e d a n d r e p l a c e d b y a m o r e satisfactory expression. Some General

Considerations

T h e basic r e s u l t i n s t a t i s t i c a l m e c h a n i c s is t h a t the p r o b a b i l i t y of a system b e i n g i n a state specified b y a n e n e r g y E{ is p r o p o r t i o n a l to t h e B o l t z m a n n f a c t o r exp{ —

where p =

d y n a m i c p r o p e r t i e s m a y be specified.

1/kT.

W i t h this result, t h e r m o -

F o r example, the thermodynamic

e n e r g y is

i

U

d In Zjv

(1)

where 'N

exp{-/M} £

exp {—

pEJ


(2)

1.

Fluids

H E N D E R S O N

and Fluid

3

Mixtures

is c a l l e d t h e p a r t i t i o n f u n c t i o n a n d A is the H e l m h o l t z free e n e r g y A =

U — TS).

(i.e.,

W i t h e x c e p t i o n of a f e w fluids, s u c h as h e l i u m a n d

h y d r o g e n , the e n e r g y levels Ei f o r m a c o n t i n u u m a n d the s u m i n E q u a t i o n 2 c a n be r e p l a c e d b y a n i n t e g r a l . T h u s , for a system of N m o l e c u l e s

Z n

=

Tfiflf

e

x

t-i ^/) *

p

V*

dp

8

d

( 3 )

Downloaded by NORTH CAROLINA STATE UNIV on August 22, 2013 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch001

w h e r e the ps a n d q's are t h e g e n e r a l i z e d m o m e n t a a n d coordinates, s is the n u m b e r of degrees of f r e e d o m , a n d system.

T h e factor h

8

is the H a m i l t o n i a n of

the

arises f r o m the v o l u m e associated w i t h q u a n t u m

states i n phase space a n d the factor N\ appears because the molecules i n a fluid are i n d i s t i n g u i s h a b l e . G e n e r a l l y the m o l e c u l e s of the fluid w i l l h a v e i n t e r n a l degrees of freedom.

If these i n t e r n a l degrees of f r e e d o m

are i n d e p e n d e n t of t h e

d e n s i t y of the fluid (as is often t h e case) t h e y m a k e no c o n t r i b u t i o n to t h e e q u a t i o n of state a n d c a n b e i g n o r e d .

S i n c e I a m interested o n l y i n

p r e s e n t i n g g e n e r a l p r i n c i p l e s , I w i l l i g n o r e the c o n t r i b u t i o n of i n t e r n a l degrees of f r e e d o m a n d assume t h a t

i-l w h e r e $ is the p o t e n t i a l energy of the m o l e c u l e s a n d d e p e n d s o n l y u p o n the positions of the center of mass. T h e t h e o r y of fluids i n w h i c h i n t e r n a l degrees of f r e e d o m development.

Z N

where A =

c o n t r i b u t e to the e q u a t i o n of state is s t i l l

under

U s i n g E q u a t i o n 4 , the p a r t i t i o n f u n c t i o n b e c o m e s

=

~NY I

e

x

p

(""^)

d r i

•••

W

d r N

h/(2irmkT) . 1/2

T o m a k e f u r t h e r progress the f o r m of m u s t b e specified. G e n e r a l l y the p o t e n t i a l e n e r g y w i l l c o n t a i n terms i n v o l v i n g the coordinates o f p a i r s , t r i p l e t s , q u a d r u p l e t s , etc., of m o l e c u l e s .

A f e w c a l c u l a t i o n s of the e q u a -

t i o n of state of a fluid w i t h s u c h a g e n e r a l f o r m for h a v e b e e n

made.

H o w e v e r , the c o m m o n p r a c t i c e is to assume p a i r - w i s e a d d i t i v i t y : *(ri,

I n this case, u(r)

. . . , r) N

— 1 3 u(r ) i?)

V 2

^ 1728(11/2 - 8 e - + 3 e * )

(23)

1 9 c

^

_ ,

)

i

M

.

( 2 4 )

and 1 1 ^

°

3

3 5

e

"

[-

W 2 4 )

+ 2z + 4 ( l + * K * ]

E q u a t i o n 21 gives the exact r e s u l t for t>i i n t h e M S A . T h e for v , 2

v

Sy

a n d t;

are p a r a m e t r i z a t i o n s .

4

(25)

2

4

expressions

E q u a t i o n 21 is a n a l y t i c a l a n d

q u i t e easy to use. F o r most cases, v , t ; , a n d tf are s m a l l w h e n 2

3

4

compared

w i t h Vx so that i t is best to use E q u a t i o n 21 for Vx r a t h e r t h a n some approximation.

H o w e v e r , i f one wishes, s i m p l i f i c a t i o n s c a n b e

f r o m a n e x p a n s i o n of E q u a t i o n 21 i n p o w e r s of p a n d z.

obtained

One possibility

is

"

1

"

2

4

^ - ? - L

1

+ 1 0 ( l + , ) ( 7 + 2,)

(26)

\]

E x c e p t for z —> oo, most of t h e c o n t r i b u t i o n to Vi comes f r o m t h e term.

The limit z - »

oo is m a i n l y of m a t h e m a t i c a l interest.

first

I n most

situations of p h y s i c a l interest z is s m a l l . W e refer to t h e l i t e r a t u r e ( I ) the v a r i o u s i n t e g r a l equations

for a d i s c u s s i o n of t h e d e r i v a t i o n of

a n d f o r details r e g a r d i n g t h e i r s o l u t i o n

( u s u a l l y n u m e r i c a l ) for other cases.

Perturbation

Theory

P e r t u r b a t i o n t h e o r y is t h e oldest of t h e three m e t h o d s . W e w i l l see t h a t i t dates b a c k to v a n d e r W a a l s . H o w e v e r , its u t i l i t y w a s n o t a p p r e c i a t e d b y theorists u n t i l t h e last d e c a d e . I n p e r t u r b a t i o n t h e o r y w e assume t h a t w e h a v e f u l l k n o w l e d g e of s o m e reference system ( o r u n p e r t u r b e d s y s t e m ) w h o s e p r o p e r t i e s w e w i l l d e n o t e b y a s u b s c r i p t 0. W e m a y h a v e o b t a i n e d this k n o w l e d g e b y means


1.

H E N D E R S O N

Fluids

of some c o m p u t e r equation. fluid,

and Fluid

9

Mixtures

s i m u l a t i o n s or f r o m the s o l u t i o n of

some i n t e g r a l

U s u a l l y , this reference system is t a k e n to b e the h a r d - s p h e r e

where r d

/ oo,

=

u (r)=)°:> 0

lO,

(27)

W e assume that, to some a p p r o x i m a t i o n , the p a i r p o t e n t i a l is u (r)

and a

0


s m a l l p e r t u r b a t i o n . T h e simplest case is u(r)

=

+

u (r) 0

(28)

€w(r)

If the free energy is e x p a n d e d i n p o w e r s of c, w e h a v e

Az_^5. _ NkT

£

(j8c) AJNkT

(29)

n

£i

where Ai/NkT-^p and go(r)

jw(r)g (r)

is the R D F of the reference

(30)

dr

0

system.

T h e higher-order

A

n

i n v o l v e s integrals o v e r h i g h e r - o r d e r d i s t r i b u t i o n f u n c t i o n s . I f the reference system is the h a r d - s p h e r e system, A l a t e d f r o m E q u a t i o n 14.

Carnahan and Starling (8)

0

m a y be calcu-

have proposed

slight m o d i f i c a t i o n of E q u a t i o n 14 w h i c h is s l i g h t l y m o r e accurate. the C a r n a h a n a n d S t a r l i n g expression is c e r t a i n l y r e c o m m e n d e d ;

a

Using

however,

I w i s h to g i v e t h e f o l l o w i n g w a r n i n g . T h e a n a l o g u e of the C a r n a h a n a n d S t a r l i n g e q u a t i o n of state becomes v e r y i n a c c u r a t e for a m i x t u r e of h a r d spheres w h e n one of the components i n the m i x t u r e is v e r y l a r g e w h e r e a s the a n a l o g u e of E q u a t i o n 14 r e m a i n s accurate. I n as m u c h as t h e C a r n a h a n a n d S t a r l i n g - t y p e expression is i n better agreement w i t h

computer

s i m u l a t i o n s for h a r d - s p h e r e m i x t u r e s for d i a m e t e r ratios at least as l a r g e as 3 : 1 , i t is p r o b a b l e that this d e f i c i e n c y p r o b a b l y is i r r e l e v a n t to a n y p r a c t i c a l c a l c u l a t i o n . H o w e v e r , one s h o u l d be w a r y n o t o n l y to a v o i d a p p l i c a t i o n of the C a r n a h a n a n d S t a r l i n g - t y p e expression for

extremely

large d i a m e t e r ratios b u t to e x a m i n e c a r e f u l l y a n y p r e d i c t i o n s b a s e d o n the use of this expression i n situations i n w h i c h i t has n o t b e e n s t u d i e d i n d e t a i l . F o r a h a r d - s p h e r e reference

fluid,

go(0

a n d thus A i , c a n b e

o b t a i n e d either f r o m the P Y h a r d - s p h e r e results ( 5 ) simulations ( 9 , 1 0 ) .

or from

computer

F o r most cases, A i m u s t be o b t a i n e d b y n u m e r i c a l

integration. H o w e v e r , if w(r)

=

—c exp {—z(x

—

l)}/x


(31)

10

EQUATIONS

where x = yield

r/a- a n d the P Y g o ( r )

>

A

N

k

T

O F

S T A T E

are u s e d , E q u a t i o n 16 m a y b e u s e d to

- - e W % z )

( 3 2 )

T h e s i m i l a r i t y to E q u a t i o n 21 is not a c c i d e n t a l . I f a h a r d - s p h e r e reference

fluid

is u s e d , the s e c o n d - o r d e r

t e r m has


the f o r m A /NkT 2

=

r*[w(rd)

] g (r)dr

w(rid)w(r d)

F (r r )

-TrptP J°°

+ff

2

2

0

0

l9

(33)

dr dr

2

x

2

B a r k e r a n d H e n d e r s o n ( J O ) h a v e g i v e n a c o n v e n i e n t p a r a m e t r i z a t i o n of Fo(r

for the h a r d - s p h e r e reference

r)

u

2

fluid.

So f a r a l l w e h a v e is f o r m a l i s m . O n e c o u l d a r g u e t h a t t h e r e is n o reason to b e l i e v e that E q u a t i o n 29 is u s e f u l e x c e p t at h i g h t e m p e r a t u r e s , w h e r e /?c is s m a l l . T h e u t i l i t y of p e r t u r b a t i o n t h e o r y e v e n at t e m p e r a t u r e s as l o w as t h e t r i p l e p o i n t , w a s first p o i n t e d o u t a d e c a d e ago b y B a r k e r a n d H e n d e r s o n (11)

w h o a r g u e d t h a t the r e l e v a n t p a r a m e t e r i n deter-

m i n i n g the c o n v e r g e n c e of E q u a t i o n 29 w a s not t h e smallness of /Jc b u t the smallness of the effect of t h e p e r t u r b a t i o n o n t h e s t r u c t u r e of fluid.

T h e y noted that A

x

gives t h e effect of w(r)

p r o p e r t i e s i n the absence of a n y changes i n s t r u c t u r e a n d t h a t A

2

t h e effect of changes

the

on the thermodynamic gives

i n structure. A t h i g h densities s u c h changes

in

structure are suppressed because the m o l e c u l e s are p a c k e d t i g h t l y . T h e r e fore, A

is s m a l l c o m p a r e d w i t h A i (as is o b s e r v e d b y d i r e c t c a l c u l a t i o n

2

of A i a n d A ) . T h e h i g h e r - o r d e r A

n

2

are e v e n s m a l l e r a n d , as B a r k e r a n d

H e n d e r s o n suggested, c a n be n e g l e c t e d i n most c a l c u l a t i o n s . A t l o w e r densities, p a r t i c u l a r l y n e a r the c r i t i c a l p o i n t , changes

in

structure are easier a n d the c o n v e r g e n c e of the p e r t u r b a t i o n e x p a n s i o n is slower.

B u t e v e n there, second-order

p e r t u r b a t i o n t h e o r y gives

quite

r e a s o n a b l e results. A l l of this makes sense as l o n g as u(r) p l u s ew(r).

is the h a r d - s p h e r e p o t e n t i a l

Unfortunately, the potentials w h i c h occur i n real applications

are not of this f o r m . N e v e r t h e l e s s , f o l l o w i n g B a r k e r a n d H e n d e r s o n

(12),

we can write u(r)

=

Uo(r)

+

(34)

w(r)

€

w h e r e u (r) a n d w(r) are the p o s i t i v e a n d n e g a t i v e parts of u(r). Thus, w e c a n use E q u a t i o n s 29 a n d 30. H o w e v e r , w e n o w h a v e a n u n f a m i l i a r reference fluid. F o r t u n a t e l y , for most a p p l i c a t i o n s u (r) is v e r y steep 0

0


1.

H E N D E R S O N

Fluids

11

and Fluid Mixtures

a n d c a n b e r e p l a c e d b y t h e h a r d - s p h e r e p o t e n t i a l i f t h e d i a m e t e r d is chosen j u d i c i o u s l y . B y means of a f o r m a l e x p a n s i o n i n p o w e r o f a n i n v e r s e steepness p a r a m e t e r , B a r k e r a n d H e n d e r s o n s h o w e d t h a t

d — f

a

[1 -

exp {-pu«(z)}]dz

(35)

J o

w h e r e o- is the p o i n t at w h i c h u(r) changes s i g n . W i t h t h e a b o v e c h o i c e


of d, a u s e f u l s e c o n d - o r d e r p e r t u r b a t i o n t h e o r y is o b t a i n e d f r o m E q u a tions 14, 29, 30, a n d 32. T h e results of this second-order p e r t u r b a t i o n t h e o r y f o r a fluid w h o s e p a i r p o t e n t i a l is the L e n n a r d - J o n e s 6:12 p o t e n t i a l ,

w(r)

=

Equations of State in Engineering and Research - American Chemical

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